基于局部多项式拟合非时齐扩散模型参数的局部估计

摘要

This paper studies local estimations of parameters for a class of time-inhomogeneous diffusion models, which is essential building blocks for option pricing and risk man-agement. The parameters in our models are the key variables in option pricing and risk management. Based on discretely observed samples, we propose the local polynomial composite quantile regression (CQR) estimation of drift parameters by using local polynomial fitting, and verify its asymptotic properties. Considering dif-fusion parameter being positive, we take it to be locally log-polynomial fitting and obtain its kernel weighted estimation. The asymptotic bias, asymptotic variance and asymptotic normality of the estimation for volatility function are discussed. Separate bandwidths are used for the estimations of drift and diffusion parameters. Simulation studies show that the proposed local CQR estimations perform better than the local least squares estimations.%本文主要研究一类非时齐扩散模型中参数的局部估计，此类问题是期权定价和风险管理中必要的组成部分。漂移参数和扩散参数是期权定价和风险管理中的关键性变量。首先，基于离散观测样本，利用局部多项式拟合，得到了漂移参数的局部多项式复合分位回归估计，并证明了其渐近性质。然后，考虑到扩散参数是非负的，本文利用对数局部多项式拟合，得到了扩散参数的局部多项式估计，并讨论了扩散项估计的渐近偏差、渐近方差和渐近正态性。最后，分别对漂移参数和扩散参数的估计采用了不同的带宽参数。模拟结果表明，本文所得到的局部复合分位回归估计比局部最小二乘估计的拟合效果更好。